Abstracts
Inthisnote, we obtain a Chung's integral test for self-normalized sums of i.i.d. random variables. Furthermore, we obtain a convergence rate of Chung law of the iterated logarithm for self-normalized sums.
Chung's integral test; self-normalized sums; convergence rate
Nesta nota, obtemos um teste integral de Chung para somas auto-normalizadas de variáveis aleatórias i.i.d. (independentes e identicamente distribuídas). Além disso, obtemos uma taxa de convergência da lei de Chung do logaritmo iterado para somas auto-normalizadas.
teste integral de Chung; somas auto-normalizadas; taxa de convergência
MATHEMATICAL SCIENCES
On the other law of the iterated logarithm for self-normalized sums
Guang-Hui Cai
Department of Mathematics and Statistics, Zhejiang Gongshang University, Hangzhou 310035, P.R. China
ABSTRACT
Inthisnote, we obtain a Chung's integral test for self-normalized sums of i.i.d. random variables. Furthermore, we obtain a convergence rate of Chung law of the iterated logarithm for self-normalized sums.
Key words: Chung's integral test, self-normalized sums, convergence rate.
RESUMO
Nesta nota, obtemos um teste integral de Chung para somas auto-normalizadas de variáveis aleatórias i.i.d. (independentes e identicamente distribuídas). Além disso, obtemos uma taxa de convergência da lei de Chung do logaritmo iterado para somas auto-normalizadas.
Palavras-chave: teste integral de Chung, somas auto-normalizadas, taxa de convergência.
1 INTRODUCTION
Let X, X1, X2,... be i.i.d. random variables with mean zero and variance one, and set
Also let logx = ln(x ∨ e), log2x = log(logx). Then by the so-called Chung's law of the iterated logarithm we have
This result was first proved by Chung (1948) under E|X|3 < ∞, and by Jain and Pruitt (1975) under the sole assumption of a finite second moment. Einmahl (1989) obtained the Darling Erdös theorem for sums of i.i.d. random variables. Griffin and Kuelbs (1989) got Self-normalized laws of the iterated logarithm. Griffin and Kuelbs (1991) obtained some extensions of the laws of the iterated logarithm via self-normalized.Lin (1996) got a self-normalized Chung-type law of iterated logarithm. Einmahl (1993) obtained the following integral test refining (1.1) under the minimal conditions.
THEOREM A. Let {X, Xn; n > 1} be a sequence of i.i.d. random variables with EX = 0, EX2 = 1 and
Then for any eventually non-decreasing function Ø:[1, ∞) → (0, ∞),
Einmahl (1993) showed that if (1.2) is not true, Theorem A is false. We thus see that condition (1.2) is sharp. However, if we use Vn to replace , we can eliminate the condition (1.2) in Theorem A. Explicitly, we get the following theorem.
THEOREM 1.1. Let {X, Xn; n > 1} be a sequence of i.i.d. random variables with EX = 0, EX2 = 1. Then for any eventually non-decreasing function Ø:[1, ∞) → (0, ∞),
Our next theorem gives a result on a convergence rate of (1.1).
THEOREM 1.2. Let {X, Xn; n > 1} be a sequence of i.i.d. random variables with EX = 0, EX2 = 1. Then for any b > -1, we have
Throughout this note, let C denote a positive constant, whose values can differ in different places.
2 PROOF
PROOF OF THEOREM 1.1. It is enough to prove the result for eventually non-decreasing function Ø:[1, ∞) → (0, ∞) satisfying
(See Einmahl 1993). Let
Observe that by (2.1)
where Ψ(t) = Ø(t)3/(1+Ø(t)2), t > 1, and similarly,
where Ψ'(t) = Ø(t)3/(Ø(t)2 -1), t > 1. It is easily checked that J(Ø) < ∞ implies J(Ψ) < ∞ and J(Ø) = ∞ implies J(Ψ') = ∞, and by Theorem 1 of Einmahl (1993), we have
J(Ψ) < ∞ ⇒ P (Mn < Bn / Ψ(n) i.o. = 0
and
J(Ψ') = ∞ ⇒ P (Mn < Bn / Ψ'(n) i.o. = 0
Now by Lemma 2.2 below,
P(Mn< Vn / Ø(n), Δn>(log2n)-3/2 i.o.) = 0
and
P(Mn< Bn / Ø(n), Δn>(log2n)-3/2 i.o.) = 0.
From these equations and (2.2), (2.3), hence we see that Theorem 1.1 holds true.
We now present two lemmas used in the main proof of Theorem 1.1.
LEMMA 2.1. For any x > 0 there exist positive constants η = η(x) and A = A(x) such that
PROOF. See the Lemma 2(b) of Einmahl (1993).
LEMMA 2.2. We have
and
PROOF. Let
and
First using EX2 = 1, we have
and
Thus, it follows by applying Corollary 3.1 of Lin et al. (1999, P.95) and Borel-Cantelli lemma that
Using strong law of large numbers and Hartman-Wintner LIL, we have
Thus, by EX2 = 1, we obtain that for large n,
Recalling that
< n, n > 1 and limn→ ∞ /n = 1 a.s., in order to prove (2.4) and (2.5), by (2.1), (2.6) and (2.7), it suffices to show that
Now, set m(n):= [n/(log2n)9], n > 1. By EX2 = 1, we have
Applying Kolmogorov's LIL and EX2 = 1, we have
it easily follows from above inequalities that
Hence observe that on account of (2.9) it is enough to show that
Let nk = 2k and mk = [2k/(logk)10], k > 0, for large enough k,
Thus, in order to prove (2.10), it suffices to show that
Let
Notice that
Mnk-1, j<Mnk-1+|Xj| < 3 Mnk-1, 1 < j < nk-1.
Using the independence and Lemma 2.1, it is clear that for some constant η > 0 and large enough k,
Finally, By Lemma 4 of Einmahl (1993), we have
and hence we obtain (2.11) from the Borel-Cantelli lemma.
PROOF OF THEOREM 1.2. For each n > 1 and 1 < i < n, we have
By EX2 = 1, it is easy to show that < n. Hence for some < δ < 1 and any ε > 0
In order to prove (1.5), it suffices to show that for any b > -1
By Lemma 2.1, there exists a positive constant η such that
Since EX = 0 and EX2 = 1, there exists a positive integer n0 such that for all n > n0
Hence using the Bernstein inequality, there exists a positive constant β < 1/3000 such that
Finally, by EX2 = 1, we have
Thus, (2.12) holds true.
ACKNOWLEDGMENTS
This paper was supported by National Natural Science Foundation of China and Youth Talent Foundation of Zhejiang Gongshang University (Q07-07).
Manuscript received on March 17, 2008; accepted for publication on May 15, 2008; presented by ARON SIMIS
AMS Classification: 60F15, 62F05
E-mail: cghzju@163.com
- CHUNG KL. 1948. On the maximal partial sums of sequences of independent random variables. Trans Amer Math Soc 4: 205233.
- EINMAHL U. 1989. The Darling Erdös theorem for sums of i.i.d. random variables. Probab Theory Relat Fields 82: 241257.
- EINMAHL U. 1993. On the other law of the iterated logarithm. Probab Theory Relat Fields 96: 97106.
- GRIFFIN PS AND KUELBS JD. 1989. Self-normalized laws of the iterated logarithm. Ann Probab 17: 15711601.
- GRIFFIN PS AND KUELBS JD. 1991. Some extensions of the laws of the iterated logarithm via self-normalized. Ann Probab 19: 380395.
- JAIN NC AND PRUITT WE. 1975. The other law of the iterated logarithm. Ann Probab 3: 10461049.
- LIN ZY. 1996. A self-normalized Chung-type law of iterated logarithm. Theory Probab Appl 41: 791798.
- LIN ZY, LU CR AND SU AG. 1999. Limit Theoretical Basis of Probability. Hight Education Press, Beijing.
Publication Dates
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Publication in this collection
28 Aug 2008 -
Date of issue
Sept 2008
History
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Accepted
15 May 2008 -
Received
17 Mar 2008